Statistical Process Control PDF

Statistical Process Control  Learning Objectives § Explain the purpose of a control chart § Explain the role of the central limit theorem in SPC § Build 𝑥-charts ̅ and 𝑅-charts § List the five steps involved in building control charts § Build 𝑝-charts and 𝑐-charts...

Statistical Process Control  Learning Objectives § Explain the purpose of a control chart § Explain the role of the central limit theorem in SPC § Build 𝑥-charts ̅ and 𝑅-charts § List the five steps involved in building control charts § Build 𝑝-charts and 𝑐-charts § Explain acceptance sampling  Statistical Process Control § Application of statistical techniques to ensure that processes meet standards. § Provide a statistical signal when assignable (special) causes of variation are present § Eliminate assignable causes of variation  Statistical Process Control Variability is inherent in every process i. Natural or common causes ii. Special or assignable causes  Variations Natural Variations Assignable Causes § Common causes that affect § Indicates the presence of all processes changes in the process § Expected variations § Variations can be traced to a specific reason - misadjusted § Output measures follow a equipment, machine wear, probability distribution fatigue, untrained workers, new § Process “in control” if the raw material distribution of output falls § Eliminate bad causes, and within acceptable limits incorporate new causes  Key Terms Terminology Explanation Population Entire group or set of items under study Sample Subset of the population selected for the study Sampling Process of selecting a subset of data points from larger population for analysis Sample size Number of observations included in a sample Distribution Systematic way to describe the likelihood of different outcomes in a random process Normal Symmetric, bell-shaped probability distribution distribution characterized by mean and standard deviation Sampling Distribution of a statistics (e.g. mean) calculated distribution from multiple samples of the same size  Key Terms Terminology Explanation Dispersion Extent to which dataset spread out from the central tendency Central tendency Center of a distribution – commonly measured by mean, median or mode Mean Average of a set of values Standard A measure of the typical deviation of values from deviation the mean  Samples To measure the process, we take samples and analyze the sample statistics following these steps Samples of the product, say five After enough samples are taken boxes of cereal taken off the filling from a stable process, they form machine line, vary from each other a pattern called a distribution in weight Each of these represents one sample of five boxes of cereal The solid line represents the distribution Frequency # # Frequency # # # # # # # # # # # # # # # # # # # # # # # # Weight Weight  Samples There are many types of distributions, including the normal (bell- shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Central tendency Variation Shape Frequency Weight Weight Weight  Samples If only natural causes of If assignable causes are variation are present, the output present, the process output is of a process forms a distribution not stable over time and is not that is stable over time and is predictable predictable ? ?? ?? ? ? ? ? ? ? ? ? ??? ??? Frequency Frequency Prediction Prediction e e T im T im Weight Weight  Key Terms Terminology Explanation Variation Natural and special differences in a process Control limits Upper and lower bounds on a control chart that define the acceptable range of variation Centerline Represents the average or mean of the process – baseline for accessing the stability Out of control Points of control chart fall outside the control limits – presence of special cause variation – corrective action may be needed In control All the points on a control chart fall within the control limit – process is stable and operating as expected  Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes  Process Control (a) In statistical control and capable of producing Frequency within control limits Lower control limit Upper control limit (b) In statistical control but not capable of producing within control limits (c) Out of control Size (weight, length, speed, etc.)  Central Limit Theorem Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve 1) The mean of the sampling distribution will be the same as the population mean µ x= = µ 2) The standard deviation of the sampling distribution (𝜎#̅ ) will equal the population σ standard deviation (s ) divided by the σx = square root of the sample size, n n Population and Sampling Distributions Population Distribution of sample means distributions always approaches a normal distribution = Beta Mean of sample means = x Standard deviation of σ the sample =σx = Normal n means Uniform | | | | | | | −3σ x −2σ x −1σ x x= +1σ x +2σ x +3σ x 95.45% fall within ± 2σ x 99.73% of all x fall within ± 3σ x  Sampling Distribution Sampling distribution of means Process distribution from Process distribution which the sample was drawn of means was also normal, but it could have been any distribution 𝑥̿ = 𝜇 (mean) n = 100 n = 50 As the sample size increases, n = 25 the sampling distribution narrows Mean of process  Types of Control Charts Known standard deviation 𝑥̅ −chart : Changes in mean Unknown Continuous standard variables deviation 𝑅 −chart: Changes in dispersion 𝑝 −chart: Fraction, proportion or percentage defects Categorical variables 𝑐 −chart: Count defects per unit output  Control Charts for Variables § Continuous random variables with real values § May be in whole or in fractional numbers % −chart 𝒙 R-chart Tracks changes in Indicates a gain or loss central tendency (mean) of dispersion Due to factors such as Due to changes in worn tool wear, gradual bearings, loose tool, increase in temperature, sloppiness of operators new materials  Setting Chart Limits For 𝑥̅ −charts when we know 𝝈 Lower control limit (LCL#̅ )= 𝑥̿ − 𝑧𝜎#̅ Upper control limit (UCL#̅ )= 𝑥̿ + 𝑧𝜎#̅ where 𝑥̿ = mean of the sample means or a target value set for the process 𝑧 = number of normal standard deviations 𝜎*̅ = standard deviation of the sample means = +⁄, 𝜎 = population (process) standard deviation 𝑛 = sample size  Example: Setting Control Limits § Randomly select and weigh nine (𝑛 = 9) boxes of cereals at each hour § Managers want to set control limits that include 99.73% of the sample means (𝑧 = 3) § Population standard deviation is known to be 1 oz (𝑠 = 1) Find the average weight in the FIRST sample Example: Setting Control Limits WEIGHT OF WEIGHT OF WEIGHT OF SAMPLE SAMPLE SAMPLE (AVG. OF (AVG. OF (AVG. OF 9 HOUR 9 BOXES) HOUR 9 BOXES) HOUR BOXES) 1 16.1 5 16.5 9 16.3 2 16.8 6 16.4 10 14.8 3 15.5 7 15.2 11 14.2 4 16.5 8 16.4 12 17.3 " 12 % x= = 16 ounces $ ( ∑ Avg of 9 boxes ) ' n=9 Average mean of = $ x= = i=1 ' 12 samples $ 12 ' z=3 $ ' # & σ = 1 ounce  Example: Setting Control Limits " % x= = 16 ounces 12 ( $ ∑ Avg of 9 boxes Average mean $= i=1 ) ' = x= ' n=9 of 12 samples $ 12 ' $ ' z=3 # & σ = 1 ounce  Example: Setting Control Chart Variation due for samples of Out of to assignable 9 boxes control causes Control 17 = UCL Limits 16 = Mean Variation due to natural causes 15 = LCL (AVG. OF 9 (AVG. OF 9 Variation due HOUR BOXES) HOUR BOXES) | | | | | | | | | | | | to assignable 1 2 3 4 5 6 7 8 9 10 11 12 Out of causes 1 16.1 7 15.2 Sample number control 2 16.8 8 16.4 3 15.5 9 16.3 4 16.5 10 14.8 5 16.5 11 14.2 6 16.4 12 17.3  Setting Chart Limits For 𝑥̅ −charts when we don’t know 𝝈 Lower control limit (LCL#̅ )= 𝑥̿ − 𝐴$ 𝑅9 Upper control limit (UCL#̅ )= 𝑥̿ + 𝐴$ 𝑅9 where 𝑥̿ = mean of the sample means 𝐴; = control chart factor found in Table S6.1 n R ∑ i = average range of the samples 𝑅) = R = i=1 = n  Control Chart Factors Table S6.1 Factors for Computing Control Chart Limits (3 sigma) SAMPLE SIZE, MEAN FACTOR, UPPER RANGE, LOWER RANGE, n A2 D4 D3 2 1.880 3.268 0 3 1.023 2.574 0 4.729 2.282 0 5.577 2.115 0 6.483 2.004 0 7.419 1.924 0.076 8.373 1.864 0.136 9.337 1.816 0.184 10.308 1.777 0.223 12.266 1.716 0.284 Example: Setting Control Limits using Table Values Given: Super Cola Example Process average = 12 ounces Labeled as “net weight Average range =.25 ounce 12 ounces” Sample size = 5 UCL = 12.144 Mean = 12 LCL = 11.856  𝑅-chart § Shows sample ranges over time § Difference between smallest and largest values in the sample § Monitors process variability § Independent from the process mean  Setting Chart Limits For 𝑅 −charts Lower control limit (LCL&% )= 𝐷' 𝑅% Upper control limit (UCL&% )= 𝐷( 𝑅% where 𝑈𝐶𝐿"! = upper control limit for the range 𝐿𝐶𝐿"! = lower control limit for the range 𝐷# and 𝐷$ = values from Table S6.1  Control Chart Factors Table S6.1 Factors for Computing Control Chart Limits (3 sigma) SAMPLE SIZE, MEAN FACTOR, UPPER RANGE, LOWER RANGE, n A2 D4 D3 2 1.880 3.268 0 3 1.023 2.574 0 4.729 2.282 0 5.577 2.115 0 6.483 2.004 0 7.419 1.924 0.076 8.373 1.864 0.136 9.337 1.816 0.184 10.308 1.777 0.223 12.266 1.716 0.284  Example: Setting Control Limits Given: Average range = 5.3 pounds Sample size = 5  Salmon 11.5 – x Bar Chart UCL = 11.524 Fillets at Sample Mean 11.0 – = x = – 10.959 Darden 10.5 – Restaurant | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 LCL = – 10.394 Range Chart 0.8 – UCL = 0.6943 Sample Range 0.4 – – = 0.2125 R 0.0 – | | | | | | | | | LCL = 0 1 3 5 7 9 11 13 15 17 Even though the process average is under control, the dispersion of the process may not be. Operations managers use control charts for ranges to monitor process variability and control charts for averages to monitor process central tendency  Mean and Range Charts The mean chart is sensitive to shifts in the process mean (a) These sampling (Sampling mean is distributions result shifting upward, in the charts below but range is consistent) UCL ( 𝑥-chart ̅ detects shift 𝑥-chart ̅ in central tendency) LCL UCL (𝑅-chart does not detect 𝑅-chart change in mean) LCL  Mean and Range Charts The 𝑹-chart is sensitive to shifts in the process standard deviation (b) (Sampling mean is These sampling constant, but distributions result dispersion is in the charts increasing) below UCL ( 𝑥-chart ̅ indicates no 𝑥-chart ̅ change in central tendency) LCL UCL (𝑅-chart detects increase 𝑅-chart in dispersion) LCL Setting Other Control Limits Common z Values 𝑧 −value (Standard deviation Desired control required for desired level of limits (%) confidence) 90.0 1.65 95.0 1.96 95.45 2.00 99.0 2.58 99.73 3.00  Exercise Sample Sample Mean (inch) Range (inch) 1 10.002 0.011 Twelve samples, each containing 2 10.002 0.014 five parts, were taken from a 3 9.991 0.007 process that produces steel 4 10.006 0.022 rods. The length of each rod in 5 9.997 0.013 the samples was determined. 6 9.999 0.012 The results were tabulated and 7 10.001 0.008 sample means and ranges were 8 10.005 0.013 computed. The results were: 9 9.995 0.004 10 10.001 0.011 11 10.001 0.014 12 10.006 0.009  Exercise a) Determine the upper and lower control limits and the overall means for x-bar charts and R-charts. b) Draw the charts and plot the values of the sample means and ranges. c) Do the data indicate a process that is in control? d) Why or why not?  Sample Range Sample Mean (inch) (inch) 1 10.002 0.011 2 10.002 0.014 3 9.991 0.007 4 10.006 0.022 5 9.997 0.013 6 9.999 0.012 7 10.001 0.008 8 10.005 0.013 9 9.995 0.004 10 10.001 0.011 11 10.001 0.014 12 10.006 0.009  Sample Range Sample Mean (inch) (inch) 1 10.002 0.011 2 10.002 0.014 3 9.991 0.007 4 10.006 0.022 5 9.997 0.013 6 9.999 0.012 7 10.001 0.008 8 10.005 0.013 9 9.995 0.004 10 10.001 0.011 11 10.001 0.014 12 10.006 0.009 Steps in Creating Control Charts Collect samples ) Compute overall means (𝑥̿ and 𝑅) Set appropriate control limits (𝑧 values) Calculate UCL and LCL Graph 𝑥̅ and 𝑅 charts Investigate patterns Identify and address assignable causes Revalidate with new data  Control Charts for Variables § Categorical variables – defective/non-defective, good/bad, yes/no, acceptable/non-acceptable § Measurement is typically counting defects 𝒑 −chart 𝒄 −chart Percent defective Number of defects Requires a sample size Does not require sample size information  Control Limits for p–charts Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics σ p is estimated by UCL p = p + zσ p ( p 1− p ) LCL p = p − zσ p σ̂ p = n where p = mean fraction (percent) defective in the samples z = number of standard deviations σ p = standard deviation of the sampling distribution n = number of observations in each sample  Example: 𝑝–Charts for Data Entry CEO wants to set control limit to include 99.73% of the random variation in the data entry process. He examines 100 records entered and counts the number of errors and the fraction defective in each samples. Samples of the work of 20 clerks are as follows: SAMPLE NUMBER OF FRACTION SAMPLE NUMBER OF FRACTION NUMBER ERRORS DEFECTIVE NUMBER ERRORS DEFECTIVE 1 6.06 11 6.06 2 5.05 12 1.01 3 0.00 13 8.08 4 1.01 14 7.07 5 4.04 15 5.05 6 2.02 16 4.04 7 5.05 17 11.11 8 3.03 18 3.03 9 3.03 19 0.00 10 2.02 20 4.04 80  Example: p-Chart for Data Entry NUMBER NUMBER SAMPLE OF FRACTION SAMPLE OF FRACTION NUMBER ERRORS DEFECTIVE NUMBER ERRORS DEFECTIVE 1 6.06 11 6.06 2 5.05 12 1.01 3 0.00 13 8.08 4 1.01 14 7.07 5 4.04 15 5.05 6 2.02 16 4.04 7 5.05 17 11.11 8 3.03 18 3.03 9 3.03 19 0.00 10 2.02 20 4.04 80 UCL p = p + zσ p LCL p = p − zσ p  Example: p-Chart for Data Entry.11 –.10 – UCLp = 0.10.09 – Fraction defective.08 –.07 –.06 –.05 –.04 – p = 0.04.03 –.02 –.01 – LCLp = 0.00 | | | | | | | | | |.00 – 2 4 6 8 10 12 14 16 18 20 Sample number  Example: p-Chart for Data Entry Possible assignable causes present.11 –.10 – UCLp = 0.10.09 – Fraction defective.08 –.07 –.06 –.05 –.04 – p = 0.04.03 –.02 –.01 – LCLp = 0.00 | | | | | | | | | |.00 – 2 4 6 8 10 12 14 16 18 20 Sample number  Control Limits for c-Charts Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics Lower control limit (LCL) )= 𝑐̅ − 3 𝑐̅ Upper control limit (UCL) )= 𝑐̅ + 3 𝑐̅ where 𝑐̅ = mean number of defects per unit 𝑐̅ = standard deviation of defects per unit  Example: c-Chart for Cab Company The company receives complaints everyday, and over a 9-day period, the owner received the following number of calls: 3, 0, 8, 9, 6, 7, 4, 9, 8 for a total of 54 complaints. It wants to compute 99.73% control limits. 14 – UCLc = 13.35 Number defective 12 – 10 – 8 – 6 – c= 6 4 – 2 – LCLc = 0 0 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Day Example: c-Chart for Cab Company 14 – UCLc = 13.35 Number defective 12 – 10 – 8 – 6 – c= 6 4 – 2 – LCLc = 0 0 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Day Managerial Issues and Control Charts Which parts are critical to success? Select points in the processes that need SPC Which parts have a tendency to become out of control? Variable chart Major Determine the monitor weights or Management appropriate dimensions Decisions charting Attribute charts are technique more of a yes-no or go-no go gauge Set clear What to do when policies and the process is out procedures of control  Patterns in Control Charts Run test – A test to examine the points in a control chart to see if nonrandom variation is present § Identify abnormalities in a process § Runs of 5 or 6 points above or below the target or centerline suggest assignable causes may be present § There are a variety of run tests UCL Target LCL Normal behavior. One plot out. Trends in either Process is “in Investigate for direction, 5 plots. control.” cause. Process is Investigate for cause “out of control.” of progressive change. UCL Target LCL Two plots very near Run of 5 above (or Erratic behavior. lower (or upper) below) central line. Investigate for control. Investigate Investigate for cause. for cause. cause.  Acceptance Sampling Form of quality testing used for incoming materials or finished goods Sampling Inspection Decision Take samples Inspect each Whether to at random of the items in reject the lot from a “lot” / the sample based on the batches of inspection items result Both attributes and variables can be inspected  Acceptance Sampling § Only screen “lots” – does not drive quality improvement efforts § Rejects “lots” can be: § Returned to supplier § To be 100% inspected to cull out all defects § May be re-graded to a lower specification Operating Characteristic Curve § Shows the relationship between the probability of accepting a lot and its quality level § The curve pertains to a specific plan – a combination of n (sample size) and c (acceptance level) AQL LTPD Acceptance Quality Level (AQL) § Poorest level of quality we are willing to accept Lot Tolerance Percent Defective (LTPD) § Quality level that we consider bad  The “Perfect” OC Curve Keep whole P(Accept Whole Shipment) shipment 100 – 75 – Return whole 50 – shipment 25 – Cut-Off 0 |– | | | | | | | | | | 0 10 20 30 40 50 60 70 80 90 100 % Defective in Lot  An OC Curve a = 0.05 producer's risk for AQL The steeper the Probability of curve, the better Acceptance the plan b = 0.10 | | | | | | | | | Percent 0 1 2 3 4 5 6 7 8 defective AQL LTPD Consumer's Good Indifference risk for LTPD lots Bad lots zone  Operating Characteristics Curve Producer § Responsible of replacing all the defects in the rejected lot § Producer’s risk (𝛼) – mistake of having a good lot rejected through sampling § Probability of rejecting a lot when the fraction defective is at or above the AQL Consumer § Lots accepted are the responsibility of the consumer § Consumer’s risk (𝛽) – mistake of accepting a bad lot overlooked through sampling § Probability of accepting a lot when fraction defective is below the LTPD  Average Outgoing Quality The percentage defective in an average lot of goods inspected through acceptance sampling 1. If a sampling plan replaces all defectives 2. If we know the true incoming percent defective for the lot We can compute the average outgoing quality (AOQ) in percent defective The maximum AOQ is the highest percent defective or the lowest average quality and is called the average outgoing quality limit (AOQL) Average Outgoing Quality (Pd)(Pa)(N – n) AOQ = N where Pd = true percent defective of the lot Pa = probability of accepting the lot N = number of items in the lot n = number of items in the sample  Recap Learning Objectives 1. Explain the purpose of a control chart Provide statistical signal when assignable causes of variation are present 2. Explain the role of the central limit theorem in SPC What does the theorem says? What does it provide? 3. Build 𝑥̅ −charts and 𝑅 −charts Know the formula, calculate the upper and lower limits, plot the graph, investigate patterns  Recap Learning Objectives 4. List the five steps involved in building control charts Collect samples, calculate overall means, graph the means and ranges, investigate patterns, collect additional samples 5. Build p-charts and c-charts Know the formula, calculate the upper and lower limit, plot the graph, investigate patterns 6. Explain acceptance sampling What is an acceptable lot and what is not, what are the risks and probability of accepting a bad lot